Properties of Inverse Trigonometric Functions

2.3 Properties of Inverse Trigonometric Functions

This section develops the algebraic relationships that hold among inverse trigonometric functions. Every result presented here is valid only within the principal value branches of the corresponding functions, and only for those values of $x$ for which both sides of the identity are defined. The textbook explicitly notes that some results may not hold for all values in the domains — the precise restrictions are given alongside each property.

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Foundational Relationship

Recall the fundamental link between a trigonometric function and its inverse:

If $y = \sin^{-1} x$, then $x = \sin y$. Conversely, if $x = \sin y$, then $y = \sin^{-1} x$.

This gives the two cancellation identities: